\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 5.0.2 - dD Geometry Kernel
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CGAL::Vector_d< Kernel > Class Template Reference
dD Geometry Kernel Reference » Kernel Objects

#include <CGAL/Kernel_d/Vector_d.h>

Definition

An instance of data type Vector_d<Kernel> is a vector of Euclidean space in dimension \( d\).

A vector \( r = (r_0,\ldots,r_{ d - 1})\) can be represented in homogeneous coordinates \( (h_0,\ldots,h_d)\) of number type RT, such that \( r_i = h_i/h_d\) which is of type FT. We call the \( r_i\)'s the Cartesian coordinates of the vector. The homogenizing coordinate \( h_d\) is positive.

This data type is meant for use in computational geometry. It realizes free vectors as opposed to position vectors (type Point_d). The main difference between position vectors and free vectors is their behavior under affine transformations, e.g., free vectors are invariant under translations.

Downward compatibility

We provide all operations of the lower dimensional interface x(), y(), z(), hx(), hy(), hz(), hw().

Implementation

Vectors are implemented by arrays of variables of type RT. All operations like creation, initialization, tests, vector arithmetic, input and output on a vector \( v\) take time \( O(v.dimension())\). coordinate access, dimension() and conversions take constant time. The space requirement of a vector is \( O(v.dimension())\).

Types

typedef unspecified_type LA
 the linear algebra layer.
 
typedef unspecified_type Cartesian_const_iterator
 a read-only iterator for the Cartesian coordinates.
 
typedef unspecified_type Homogeneous_const_iterator
 a read-only iterator for the homogeneous coordinates.
 
typedef unspecified_type Base_vector
 construction tag.
 

Creation

 Vector_d ()
 introduces a variable v of type Vector_d<Kernel>.
 
 Vector_d (int d, Null_vector)
 introduces the zero vector v of type Vector_d<Kernel> in \( d\)-dimensional space. More...
 
template<class InputIterator >
 Vector_d (int d, InputIterator first, InputIterator last)
 introduces a variable v of type Vector_d<Kernel> in dimension d. More...
 
template<class InputIterator >
 Vector_d (int d, InputIterator first, InputIterator last, RT D)
 introduces a variable v of type Vector_d<Kernel> in dimension d initialized to the vector with homogeneous coordinates as defined by H = set [first,last) and D: \( (\pm H_0, \pm H_1, \ldots, \pm H_{D-1}, \pm H_D)\). More...
 
 Vector_d (int d, Base_vector, int i)
 returns a variable v of type Vector_d<Kernel> initialized to the \( i\)-th base vector of dimension \( d\). More...
 
 Vector_d (RT x, RT y, RT w=1)
 introduces a variable v of type Vector_d<Kernel> in \( 2\)-dimensional space. More...
 
 Vector_d (RT x, RT y, RT z, RT w)
 introduces a variable v of type Vector_d<Kernel> in \( 3\)-dimensional space. More...
 

Operations

int dimension ()
 returns the dimension of v.
 
FT cartesian (int i)
 returns the \( i\)-th Cartesian coordinate of v. More...
 
FT operator[] (int i)
 returns the \( i\)-th Cartesian coordinate of v. More...
 
RT homogeneous (int i)
 returns the \( i\)-th homogeneous coordinate of v. More...
 
FT squared_length ()
 returns the square of the length of v.
 
Cartesian_const_iterator cartesian_begin ()
 returns an iterator pointing to the zeroth Cartesian coordinate of v.
 
Cartesian_const_iterator cartesian_end ()
 returns an iterator pointing beyond the last Cartesian coordinate of v.
 
Homogeneous_const_iterator homogeneous_begin ()
 returns an iterator pointing to the zeroth homogeneous coordinate of v.
 
Homogeneous_const_iterator homogeneous_end ()
 returns an iterator pointing beyond the last homogeneous coordinate of v.
 
Direction_d< Kernel > direction ()
 returns the direction of v.
 
Vector_d< Kernel > transform (const Aff_transformation_d< Kernel > &t)
 returns \( t(v)\).
 

Arithmetic Operators, Tests and IO

Vector_d< Kernel > & operator*= (const RT &n)
 multiplies all Cartesian coordinates by n.
 
Vector_d< Kernel > & operator*= (const FT &r)
 multiplies all Cartesian coordinates by r.
 
Vector_d< Kernel > operator/ (const RT &n)
 returns the vector with Cartesian coordinates \( v_i/n, 0 \leq i < d\).
 
Vector_d< Kernel > operator/ (const FT &r)
 returns the vector with Cartesian coordinates \( v_i/r, 0 \leq i < d\).
 
Vector_d< Kernel > & operator/= (const RT &n)
 divides all Cartesian coordinates by n.
 
Vector_d< Kernel > & operator/= (const FT &r)
 divides all Cartesian coordinates by r.
 
FT operator* (const Vector_d< Kernel > &w)
 inner product, i.e., \( \sum_{ 0 \le i < d } v_i w_i\), where \( v_i\) and \( w_i\) are the Cartesian coordinates of \( v\) and \( w\) respectively.
 
Vector_d< Kernel > operator+ (const Vector_d< Kernel > &w)
 returns the vector with Cartesian coordinates \( v_i+w_i, 0 \leq i < d\).
 
Vector_d< Kernel > & operator+= (const Vector_d< Kernel > &w)
 addition plus assignment.
 
Vector_d< Kernel > operator- (const Vector_d< Kernel > &w)
 returns the vector with Cartesian coordinates \( v_i-w_i, 0 \leq i < d\).
 
Vector_d< Kernel > & operator-= (const Vector_d< Kernel > &w)
 subtraction plus assignment.
 
Vector_d< Kernel > operator- ()
 returns the vector in opposite direction.
 
bool is_zero ()
 returns true if v is the zero vector.
 
Vector_d< Kernel > operator* (const RT &n, const Vector_d< Kernel > &v)
 returns the vector with Cartesian coordinates \( n v_i\).
 
Vector_d< Kernel > operator* (const FT &r, const Vector_d< Kernel > &v)
 returns the vector with Cartesian coordinates \( r v_i, 0 \leq i < d\).
 

Constructor & Destructor Documentation

◆ Vector_d() [1/6]

template<typename Kernel >
CGAL::Vector_d< Kernel >::Vector_d ( int  d,
Null_vector   
)

introduces the zero vector v of type Vector_d<Kernel> in \( d\)-dimensional space.

For the creation flag CGAL::NULL_VECTOR can be used.

◆ Vector_d() [2/6]

template<typename Kernel >
template<class InputIterator >
CGAL::Vector_d< Kernel >::Vector_d ( int  d,
InputIterator  first,
InputIterator  last 
)

introduces a variable v of type Vector_d<Kernel> in dimension d.

If size [first,last) == d this creates a vector with Cartesian coordinates set [first,last). If size [first,last) == p+1 the range specifies the homogeneous coordinates \( H = set [first,last) = (\pm h_0, \pm h_1, \ldots, \pm h_d)\) where the sign chosen is the sign of \( h_d\).

Precondition
d is nonnegative, [first,last) has d or d+1 elements where the last has to be non-zero.
Template Parameters
InputIteratorhas RT as value type.

◆ Vector_d() [3/6]

template<typename Kernel >
template<class InputIterator >
CGAL::Vector_d< Kernel >::Vector_d ( int  d,
InputIterator  first,
InputIterator  last,
RT  D 
)

introduces a variable v of type Vector_d<Kernel> in dimension d initialized to the vector with homogeneous coordinates as defined by H = set [first,last) and D: \( (\pm H_0, \pm H_1, \ldots, \pm H_{D-1}, \pm H_D)\).

The sign chosen is the sign of \( D\).

Precondition
D is non-zero, the iterator range defines a \( d\)-tuple of RT.
Template Parameters
InputIteratorhas RT as value type.

◆ Vector_d() [4/6]

template<typename Kernel >
CGAL::Vector_d< Kernel >::Vector_d ( int  d,
Base_vector  ,
int  i 
)

returns a variable v of type Vector_d<Kernel> initialized to the \( i\)-th base vector of dimension \( d\).

Precondition
\( 0 \leq i < d\).

◆ Vector_d() [5/6]

template<typename Kernel >
CGAL::Vector_d< Kernel >::Vector_d ( RT  x,
RT  y,
RT  w = 1 
)

introduces a variable v of type Vector_d<Kernel> in \( 2\)-dimensional space.

Precondition
\( w \neq0\).

◆ Vector_d() [6/6]

template<typename Kernel >
CGAL::Vector_d< Kernel >::Vector_d ( RT  x,
RT  y,
RT  z,
RT  w 
)

introduces a variable v of type Vector_d<Kernel> in \( 3\)-dimensional space.

Precondition
\( w \neq0\).

Member Function Documentation

◆ cartesian()

template<typename Kernel >
FT CGAL::Vector_d< Kernel >::cartesian ( int  i)

returns the \( i\)-th Cartesian coordinate of v.

Precondition
\( 0 \leq i < d\).

◆ homogeneous()

template<typename Kernel >
RT CGAL::Vector_d< Kernel >::homogeneous ( int  i)

returns the \( i\)-th homogeneous coordinate of v.

Precondition
\( 0 \leq i \leq d\).

◆ operator[]()

template<typename Kernel >
FT CGAL::Vector_d< Kernel >::operator[] ( int  i)

returns the \( i\)-th Cartesian coordinate of v.

Precondition
\( 0 \leq i < d\).